Up to a negligible factor \(n^{o(1)}\) , this gate complexity improves the best previous result of the electronic-structure simulation in the second-quantized plane-wave basis. This is because our approach improves the performance of digital quantum simulation by simultaneously exploiting commutativity of the Hamiltonian and prior knowledge of the initial state, whereas previous results were only able to employ at most one of these information. Indeed, previous work [1]} gave a Trotterization with error bound
\(\left\Vert {S}_p(t)-e^{-itH}\right\Vert _{\eta }=\mathcal {O}\left(\left(\left\Vert \tau \right\Vert +\left\Vert \nu \right\Vert _{\max }\eta \right)^{p-1}\left\Vert \tau \right\Vert \left\Vert \nu \right\Vert _{\max }\eta ^{p+2} t^{p+1}\right).\) 
